
I have extended lines from Uzbeks through the four different western Eurasians Turkic populations: Turks, Turkmen, Nogais, and Chuvashs. It is interesting that these lines intersect West Eurasians at different points:
- The line of Turks (Anatolia) with Armenians and Georgians
- The line of Turkmen (Iran) with Iranians
- The line of Nogais (North Caucasus) with Chechens
- The line of Chuvashs (far eastern Europe) beyond Russians
The only case where the fit is not almost perfect is that of the Chuvashs, who are the most northern population. It is easy enough to discover the cause of this. Chuvash have ~1.1% South Asian, as opposed to ~8.2% in the Uzbeks. There is a north-south latitudinal cline of the "South Asian" component in middle Eurasia, and the ancestors of the Chuvash moved at the northern end of that cline; of the Uzbeks at the middle.
If we add a line linking Russians with Chuvashs, we can reconstruct the hypothetical Proto-Chuvashs as a population that differed from Uzbeks in being less "South Asian", which confirms the Dodecad admixture data:
The mystery is further resolved once we look at the following PCA plot from my article On the northern/southern Caucasoid contributions to Asia 
Notice that Turks, Uzbeks and the pair of Altai and Dolgan fall along one line, while Chuvashs, Russians, and the pair of Altai/Dolgan along another.
So, the data seems consistent with the idea that the primary source of the westward Turkic expansions was something like the Altai (pics) and Dolgan, undergoing transformations and successive admixtures all the way to the Mediterranean and eastern Europe.